2x2 Graph Games
Hamburger & Guyer
-1-
What do we do?
for science: describe, formalize, predict, unify
for policy: prescribe
Game theory clarifies social, political and economic situations via
interactive decision models. To be effective at affecting policy, ...
Make it matter!
Keep it simple!
2x2 Graph Games
Hamburger & Guyer
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How can we do it?
1. Find an important situation
2. Isolate its key aspects
3. Map to a precise model
4. Expand/contract the class of models
5. Elaborate / enumerate class members
6. Find specimen situation for a model
7. Is it important? If yes, go to #2.
2x2 Graph Games
Hamburger & Guyer
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Contracting a Class of Models
an example: +/- externalities
What's the effect on Other of you taking x?
loss > x
loss < x
gain < x
gain > x
Hence the 10 Separable Games (Hamburger, 1969)
among the 78 (Rapoport and Guyer, 1966).
One of them is Prisoner's Dilemma
PD
R
T
R
0,0 1,-2
T
1,-2 -1,-1
R = refrain
T = take
2x2 Graph Games
Hamburger & Guyer
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Expanding a Class of Models
example: Take-Some, but now with n players, where
n = 3, 2x102, 3.5x107, 7x109
You take x. The total effect on all others is > x
n-PD: collective action
http://cs.gmu.edu/~henryh/npd/ (HH, 1973)
payoffs
to takers
to non-takers
number of non-takers
2x2 Graph Games
Hamburger & Guyer
Keeping it Simple
One way is: just 2 players; e.g.,
2 nations
neighboring (France, Prussia)
prominent (US, USSR)
2 economic actors
buyout
duopoly
first/second mover
labor, management
2 people
spouses
item
roommates
worker-boss
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2x2 Graph Games
Hamburger & Guyer
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Beyond the "2" & "2"
more key aspects of games




information about previous moves
knowledge about all options & consequences
computational resources
3rd player: unaligned, 3rd party, white knight, in-law
 time!
2x2 Graph Games
Hamburger & Guyer
-7-
Time, a key aspect of games,
is modeled better by graphs than by trees or matrices.
 trees imply staged steps
 matrices imply simultaneous blind moves
Graph games for aspects of time





a starting point
simultaneous possible moves
default no-action outcome
growth and decay of utility (augmented GG's)
deadlines (augmented GG's)
2x2 Graph Games
Hamburger & Guyer
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2x2 Graph Games
Hamburger & Guyer
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Chicken & Deadlines
chicken - in the news, but not analyzed
confrontation with nukes and missiles
status quo ok and stable
defend missiles not cities?
no deadline
shutdown crisis
starts with clock ticking (debt rising)
spontaneous shift at deadline (ceiling)
other deadlines
sunset clauses
election date
quarterly report
kids & cars
2x2 Graph Games
Hamburger & Guyer
-10-
3 Chicken Models
Matrix Chicken
3, 3 2, 4
4, 2 1, 1
Graph Chicken, with ruinous default
A
2, 3
1, 1
B
and ...
3, 2
(4  3)
2x2 Graph Games
Hamburger & Guyer
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Debt Ceiling Legislation as a Graph Game
If the U.S. debt ceiling is reached before congress raises it,
the government shuts down, as it did in 1995. This situation
is often called "a game of Chicken," but it's not that simple.
A
tf - t = 0
f(t)
deadline: tf
transition condition: tf - t = 0
decay function: f(t)
2x2 Graph Games
Hamburger & Guyer
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2-State 2x2 Graph Games
2 examples
Predator: 2 equivalent versions
A
A
S(2,1)
S(1,2)
T(1,2)
T(2,1)
Opposite preferences. Each can reverse them.
Thermostat
A
S(2,1)
T(1,2)
2x2 Graph Games
Hamburger & Guyer
3-state 2x2 Graph Games
Graph games that are just tree games
take this form.
S
A
T
B
U
The 6 (3!) Symmetric non-tree
3-state 2x2 GGs take this form.
A
T(,)
S(,)
B
U(,)
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2x2 Graph Games
Hamburger & Guyer
3-state 2x2 Graph Games
continued
The number of asymmetric games is
15 ( = 6C2 )
See paper for examples
The number of roles
in these games is
30 ( = 6P2 )
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2x2 Graph Games
Hamburger & Guyer
-15-
Legislating on Health
in the U.S. Senate
modeled as one of the
15 asymmetric 3-state 2x2 graph games
A
T(1,3)
S(3,2)
B
U(2,1)
Actions for each: {yield, hold}
Utilities: (politics, beliefs, danger)
A accepts "party of no" label.
B needs results; clock is ticking.
Neither wants to yield.
Other aspects:
Bill can change over time.
Mavericks are part of context.